Geodesic
Implicit Function Theorem in a Neighborhood of an Abnormal Point
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 26-33

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We study the existence of an implicit function, defined by an equation $G(x,\sigma )=0$, in a neighborhood of an abnormal point $(x_0,\sigma _0)$. We prove that if some $\lambda $-truncation of the mapping $F(x) = G(x,\sigma _0)$ is regular in a certain direction, then the sought implicit function exists. 
@article{TM_2021_315_a2,
     author = {A. V. Arutyunov and K. I. Salikhova},
     title = {Implicit {Function} {Theorem} in a {Neighborhood} of an {Abnormal} {Point}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {26--33},
     publisher = {mathdoc},
     volume = {315},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2021_315_a2/}
}
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A. V. Arutyunov; K. I. Salikhova. Implicit Function Theorem in a Neighborhood of an Abnormal Point. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Differential Games, Tome 315 (2021), pp. 26-33. http://geodesic.mathdoc.fr/item/TM_2021_315_a2/